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1.5. Chaos—Or a Butterfly Spoils Laplace’s Dream 27 tiny force on Jupiter caused by a small asteroid had a significant ef- fect on its motion, then predicting the planetary orbits would be an impossible task. In the preceding section we saw that complete, C1 vector fields do give rise to a well-posed initial value problem, so Laplace seems to be on solid ground. Nevertheless, even though the initial value problems that arise in real-world applications may be technically well-posed in the above sense, they often behave as if they were ill-posed. For a class of examples that turns up frequently—the so-called chaotic systems— predictability is only an unachievable theoretical ideal. While their short-term behavior is predictable, on longer time-scales prediction becomes, for practical purposes, impossible. This may seem para- doxical at first if we have an algorithm for predicting accurately for ten seconds, then should not repeating it with that first prediction as a new initial condition provide an accurate prediction for twenty seconds? Unfortunately, a hallmark feature of chaotic systems, called “sensitive dependence on initial conditions”, defeats this strategy. Let us consider an initial value problem dx dt = V (x), x(0) = p0 and see how things go wrong for a chaotic system when we try to compute σp 0 (t) for large t. Suppose that p1 is very close to p0, say p0 − p1 δ, and let us compare σp 1 (t) and σp 0 (t). Continuity with respect to initial conditions tells us that for δ small enough σp 1 (t) at least initially will not diverge too far from σp 1 (t). In fact, for a chaotic system, a typical behavior—when p0 is near a so-called “strange attractor”—is for σp 1 (t) to at first “track” σp 0 (t) in the sense that σp 0 (t) − σp 1 (t) initially stays nearly constant or even decreases—so in particular the motions of σp 0 (t) and σp 1 (t) are highly correlated. But then, suddenly, there will be a period during which σp 1 (t) starts to veer off in a different direction, following which σp 0 (t) − σp 1 (t) will grow exponentially fast for a while. Soon they will be far apart, and although their distance remains bounded, from that time forward their motions become completely uncorrelated. If we make δ smaller, then we can guarantee that σp 1 (t) will track σp 0 (t) for a longer period, but (and this is the essence of sensitive dependence on initial condi- tions) no matter how small we make δ, the veering away and loss of correlation will always occur. The reason this is relevant